A Variational Procedure for Optimal Design

We describe a procedure to look for the minimizer of an optimal design problem whose objective function and/or constraints have explicit dependence on the gradient of the state variable [1,2]. It is known that such problems lack, in general, optimal solutions within the class of characteristic functions, due to the non-convexity in the admissible set of designs. This is essentially a consequence of the binary nature of the design variable, a characteristic function. In addition, this feature also makes impossible to utilize any of the typical optimization algorithms for continuous variables. Homogenization theory has been the main relaxation method to define and expand the range of admissible designs to incorporate composites as structural elements when the cost functional depends on the state in a linear way. However, except for a few important cases, homogenization theory would not work when there is an explicit dependence on the gradient of the state variable [2]. We first relax ehe design problem of placing two isotropic materials to allow for certain graded materials in order to avoid the binary nature of the problem, then solve it through a variational approach by introducing a singular perturbation involving the gradient of the volume fraction of the material parameter. The procedure is demonstrated for minimizing the volume fraction of one of the two materials in a square plate.